Units: Metric Prefixes Using the Dictionary Metric Prefixes To help the SI units apply to a wide range of phenomena, the 19th General Conference on Weights and Measures in 1991 extended the list of metric prefixes so that it reaches from yotta- at 1024 (one septillion) to yocto- at 10-24 (one septillionth). Here are the metric prefixes, with their numerical equivalents stated in the American system for naming large numbers : yotta- (Y-) Notes: I am often asked about prefixes for other multiples, such as 104, 105, 10-4, and 10-5. The prefix myria- (my-) was formerly used for 104, but it is now considered obsolete and it is not accepted in the SI. To the best of my knowledge, no prefixes were ever accepted generally for 105, 10-4, or 10-5. There is a widespread misconception that prefixes for positive powers of ten are all capitalized, leading to the use of K- for kilo- and D- for deca-.

Although this does seem like a useful idea, it is not correct. **The SI Brochure spelling of this prefix is deca-, but the U.S. National Institute of Standards and Technology (NIST) recommends deka-. National variations in spelling of the prefixes are allowed by the SI . In Italian, for example, hecto- is spelled etto- and kilo- is spelled chilo-. The symbols, however, are the same in all languages, so dam (not dkm) is the symbol for the dekameter and km is the symbol for the Italian chilometro. The prefixes hecto-, deka-, deci-, and centi- are widely used in everyday life but are generally avoided in scientific work. Contrary to the belief of some scientists, however, the SI does allow use of these prefixes. The last letter of a prefix is often omitted if the first letter of the unit name is a vowel, causing the combination to be hard to pronounce otherwise. Thus 100 ares is a hectare and 1 million ohms is a megohm. However, the last letter of the prefix is not omitted if pronunciation is not a problem, as in the case of the milliampere.

The letter "l" is sometimes added to prefixes before the erg, so 1 million ergs is a megalerg (sounds odd, but better than "megerg"). Binary prefixes In computing, a custom arose of using the metric prefixes to specify powers of 2. For example, a kilobit is usually 210 = 1024 bits instead of 1000 bits. This practice leads to considerable confusion. In an effort to eliminate this confusion, in 1998 the International Electrotechnical Commission approved new prefixes for the powers of 2. These prefixes are as follows: kibi- Ei- 260 = 1 152 921 504 606 846 976 The Commission's ruling is that the metric prefixes should be used in computing just as they are used in other fields. Thus, 5 gigabytes (GB) should mean exactly 5 000 000 000 bytes, and 5 gibibytes (GiB) should mean exactly 5 368 709 120 bytes. The fate of this innovation is uncertain.

So far, very few people are using the IEC binary prefixes. Searches for them on the Internet turn up, for the most part, complaints by people who don't want to use them. Return to the Dictionary Contents page . You are welcome to email the author (rowlett@email.unc.edu) with comments and suggestions. All material in this folder is copyright &COPY; 2005 by Russ Rowlett and the University of North Carolina at Chapel Hill. Permission is granted for personal use and for use by individual teachers in conducting their own classes. All other rights reserved. You are welcome to make links to this page, but please do not copy the contents of any page in this folder to another site. The material at this site will be updated from time to time. April 16, 2005 Metric Prefixes Metric Prefixes A brief discussion of the basic metric units. In order to properly convert from one metric unit to another, you must have the prefixes memorized. One good technique is to use flashcards. Here is a search for metric prefix flashcards. There is even someone selling an e-book for metric prefix flashcards.

Note for the future: you will need to determine which of two prefixes represents a bigger amount AND you will also need to determine the exponential "distance" between two prefixes. These skills will be necessary in order to correctly convert one metric unit to another. A metric prefix is a modifier on the root word and it tells us the unit of measure. For example, centigram means we are count in steps of one one-hundredth of a gram, μg means we count by millionths of a gram. A List of the Metric Prefixes For another presentation of these prefixes, please go here . Problems concerning the units themselves There are three items - name, symbol, and size - that must be known. Problems could give any one and ask for one or both of the others. Here are only some possible problems (of many): I. Given either the name or the symbol of the prefix, give the other: 1) c 10) hecto A word to the wise: deca- (symbol = da) is a little used unit prefix. This makes it a prime target for teachers to test. Just sayin'. II.

Given the prefix size, give its name: 11) 10¯15 Go to answers for 1-15 Problems concerning the exponential distance between two prefixes This next set of problems deserves some comment. The reason is that this particular skill isn't really mentioned by chemistry (or physics) teachers. It seems that everybody just assumes students pick it up somewhere in a math class. It is an important skill that goes somewhat untaught, so I've decided to address it. The skill I'm talking about is figuring out the absolute, exponential distance between two prefixes. For example, the absolute distance between milli and centi is 101. The distance between kilo and centi is 105. What you should do is compare the two exponents as if they were placed on a number line made of exponents and the compute the absolute exponential distance between them. The key word is absolute. For example, someone might mentally do the distance between kilo and centi by comparing the exponents of positive 3 and negative 2 and getting one. So they reason the distance is 101. They would be wrong. The absolute exponential distance between 3 and -2 is 5, not 1.

Done as an exponent, the absolute exponential distance between kilo- and centi- is 105. In the problems to follow, the exponential form will be the one used. In other words, 105 is used in the solution to the problem; the 5 by itself will never be used. The 5 is only used in descriptions about how to determine the distance. Repeat: you will use the proper exponential value (like 105) in a solution to a problem; you will NEVER use just the exponent (the 5) in a solution. Here is a number line with the two prefixes in problem sixteen marked: Compute the absolute, exponential distance between two given prefixes: 16) kilo and femto SI prefixes for decimal multiples SI prefixes for decimal multiples There are 20 internationally accepted prefixes to denote 10n multiples of units.

10 of these prefixes denote (n>0) multiples, and other 10 denote (n<0) submultiples: SI Prefixes for decimal multiples Number 1 000 000 000 000 000 000 000 000 1024 1 000 000 000 000 000 000 000 1021 1 000 000 000 000 000 000 1018 1 000 000 000 000 000 1015 1 000 000 000 000 1012 0.000 000 000 000 001 10-15 0.000 000 000 000 000 001 10-18 0.000

 000 000 000 000 000 001 10-21 0.000 000 000 000 000 000 000 001 10-24 yocto y Each prefix is represented by its own symbol.

The symbols are case sensitive. Thus, m means milli and M means mega. Prefixes ranging from milli to kilo were first introduced in 1793 by the French Academy of Sciences under direction of French National Assembly as part of the Metric System. Greek prefixes were used for multiples of 10 and Latin prefixes for submultiples of 10. In 1874 BAAS adapted these prefixes as part of their CGS system, and added prefixes micro and mega. Later, 12 prefixes ranging from pico to tera were defined as part of the International System of Units — SI , which was adopted in 1960. SI is maintained by BIPM under exclusive supervision of CIPM and resolutions made by CGPM . Further 8 prefixes were added to SI in years 1964 (femto, atto), 1975 (peta, exa) and 1991 (zetta, zepto, yotta, yocto).

The prefixes have the following etymology: exa — alteration of hexa from Greek hex meaning “six” (the sixth power of 103) peta — alteration of penta from Greek pente meaning “five” (the fifth power of 103) tera — from Greek teras meaning “monster” giga — from Greek gigas meaning “giant” mega — from Greek megas meaning “great” kilo — from Greek khilioi meaning “thousand” hecto — French, alteration of Greek hekaton meaning “hundred” deca — from Greek deka meaning “ten” deci — from Latin decimus meaning tenth, from decem meaning “ten” centi — from Latin centi-, from centum meaning “hundred” milli — from Latin mille meaning “thousand” micro — from Greek mikros meaning “small” nano — from Greek nannos meaning “dwarf” pico — from Spanish pico meaning “small quantity” femto — from Danish, or Norwegian word femten meaning “fifteen” atto — from Danish, or Norwegian word atten meaning “eighteen” The names zepto and zetta are derived from septo, from

 Latin septem which means seven (the seventh power of 103) and the letter ‘z’ is substituted for the letter ‘s’ to avoid the duplicate use of the letter ‘s’ as a symbol in SI .

The names yocto and yotta are derived from Latin octo which means eight (the eighth power of 103); the letter ‘y’ is added to avoid the use of the letter ‘o’ as a symbol because it may be confused with the number zero. The CGPM has decided to name the prefixes, starting with the seventh power of 103, with the letters of the Latin alphabet, but starting from the end. Therefore the choice of letters ‘z’ and ‘y’. The initial letter ‘h’ of the word hexa in standard French is silent, so it was removed in order to simplify things. Lessons In Electric Circuits -- Volume I (DC) - Chapter 4 Lessons In Electric Circuits -- Volume I Chapter 4 Scientific notation In many disciplines of science and engineering, very large and very small numerical quantities must be managed. Some of these quantities are mind-boggling in their size, either extremely small or extremely large.

Take for example the mass of a proton, one of the constituent particles of an atom's nucleus: Proton mass = 0.00000000000000000000000167 grams Or, consider the number of electrons passing by a point in a circuit every second with a steady electric current of 1 amp: 1 amp = 6,250,000,000,000,000,000 electrons per second A lot of zeros, isn't it? Obviously, it can get quite confusing to have to handle so many zero digits in numbers such as this, even with the help of calculators and computers. Take note of those two numbers and of the relative sparsity of non-zero digits in them. For the mass of the proton, all we have is a "167" preceded by 23 zeros before the decimal point. For the number of electrons per second in 1 amp, we have "625" followed by 16 zeros.

We call the span of non-zero digits (from first to last), plus any zero digits not merely used for placeholding, the "significant digits" of any number. The significant digits in a real-world measurement are typically reflective of the accuracy of that measurement. For example, if we were to say that a car weighs 3,000 pounds, we probably don't mean that the car in question weighs exactly 3,000 pounds, but that we've rounded its weight to a value more convenient to say and remember. That rounded figure of 3,000 has only one significant digit: the "3" in front -- the zeros merely serve as placeholders. However, if we were to say that the car weighed 3,005 pounds, the fact that the weight is not rounded to the nearest thousand pounds tells us that the two zeros in the middle aren't just placeholders, but that all four digits of the number "3,005" are significant to its representative accuracy. Thus, the number "3,005" is said to have four significant figures. In like manner, numbers with many zero digits are not necessarily representative of a real-world quantity all the way to the decimal point.

When this is known to be the case, such a number can be written in a kind of mathematical "shorthand" to make it easier to deal with. This "shorthand" is called scientific notation. With scientific notation, a number is written by representing its significant digits as a quantity between 1 and 10 (or -1 and -10, for negative numbers), and the "placeholder" zeros are accounted for by a power-of-ten multiplier. For example: 1 amp = 6,250,000,000,000,000,000 electrons per second . . . can be expressed as . . . 1 amp = 6.25 x 1018 electrons per second 10 to the 18th power (1018) means 10 multiplied by itself 18 times, or a "1" followed by 18 zeros. Multiplied by 6.25, it looks like "625" followed by 16 zeros (take 6.25 and skip the decimal point 18 places to the right).

The advantages of scientific notation are obvious: the number isn't as unwieldy when written on paper, and the significant digits are plain to identify. But what about very small numbers, like the mass of the proton in grams? We can still use scientific notation, except with a negative power-of-ten instead of a positive one, to shift the decimal point to the left instead of to the right: Proton mass = 0.00000000000000000000000167 grams . . . can be expressed as . . . Proton mass = 1.67 x 10-24 grams 10 to the -24th power (10-24) means the inverse (1/x) of 10 multiplied by itself 24 times, or a "1" preceded by a decimal point and 23 zeros. Multiplied by 1.67, it looks like "167" preceded by a decimal point and 23 zeros. Just as in the case with the very large number, it is a lot easier for a human being to deal with this "shorthand" notation.

As with the prior case, the significant digits in this quantity are clearly expressed. Because the significant digits are represented "on their own," away from the power-of-ten multiplier, it is easy to show a level of precision even when the number looks round. Taking our 3,000 pound car example, we could express the rounded number of 3,000 in scientific notation as such: car weight = 3 x 103 pounds If the car actually weighed 3,005 pounds (accurate to the nearest pound) and we wanted to be able to express that full accuracy of measurement, the scientific notation figure could be written like this: car weight = 3.005 x 103 pounds However, what if the car actually did weigh 3,000 pounds, exactly (to the nearest pound)? If we were to write its weight in "normal" form (3,000 lbs), it wouldn't necessarily be clear that this number was indeed accurate to the nearest pound and not just rounded to the nearest thousand pounds, or to the nearest hundred pounds, or to the nearest ten pounds.

Scientific notation, on the other hand, allows us to show that all four digits are significant with no misunderstanding: car weight = 3.000 x 103 pounds Since there would be no point in adding extra zeros to the right of the decimal point (placeholding zeros being unnecessary with scientific notation), we know those zeros must be significant to the precision of the figure. Arithmetic with scientific notation The benefits of scientific notation do not end with ease of writing and expression of accuracy. Such notation also lends itself well to mathematical problems of multiplication and division. Let's say we wanted to know how many electrons would flow past a point in a circuit carrying 1 amp of electric current in 25 seconds.

If we know the number of electrons per second in the circuit (which we do), then all we need to do is multiply that quantity by the number of seconds (25) to arrive at an answer of total electrons: (6,250,000,000,000,000,000 electrons per second) x (25 seconds) = 156,250,000,000,000,000,000 electrons passing by in 25 seconds Using scientific notation, we can write the problem like this: (6.25 x 1018 electrons per second) x (25 seconds) If we take the "6.25" and multiply it by 25, we get 156.25. So, the answer could be written as: 156.25 x 1018 electrons However, if we want to hold to standard convention for scientific notation, we must represent the significant digits as a number between 1 and 10. In this case, we'd say "1.5625" multiplied by some power-of-ten.

To obtain 1.5625 from 156.25, we have to skip the decimal point two places to the left. To compensate for this without changing the value of the number, we have to raise our power by two notches (10 to the 20th power instead of 10 to the 18th): 1.5625 x 1020 electrons What if we wanted to see how many electrons would pass by in 3,600 seconds (1 hour)? To make our job easier, we could put the time in scientific notation as well: (6.25 x 1018 electrons per second) x (3.6 x 103 seconds) To multiply, we must take the two significant sets of digits (6.25 and 3.6) and multiply them together; and we need to take the two powers-of-ten and multiply them together. Taking 6.25 times 3.6, we get 22.5. Taking 1018 times 103, we get 1021 (exponents with common base numbers add).

So, the answer is: 22.5 x 1021 electrons . . . or more properly . . . 2.25 x 1022 electrons To illustrate how division works with scientific notation, we could figure that last problem "backwards" to find out how long it would take for that many electrons to pass by at a current of 1 amp: (2.25 x 1022 electrons) / (6.25 x 1018 electrons per second) Just as in multiplication, we can handle the significant digits and powers-of-ten in separate steps (remember that you subtract the exponents of divided powers-of-ten): (2.25 / 6.25) x (1022 / 1018) And the answer is: 0.36 x 104, or 3.6 x 103, seconds. You can see that we arrived at the same quantity of time (3600 seconds). Now, you may be wondering what the point of all this is when we have electronic calculators that can handle the math automatically.

Well, back in the days of scientists and engineers using "slide rule" analog computers, these techniques were indispensable. The "hard" arithmetic (dealing with the significant digit figures) would be performed with the slide rule while the powers-of-ten could be figured without any help at all, being nothing more than simple addition and subtraction. REVIEW: Significant digits are representative of the real-world accuracy of a number. Scientific notation is a "shorthand" method to represent very large and very small numbers in easily-handled form. When multiplying two numbers in scientific notation, you can multiply the two significant digit figures and arrive at a power-of-ten by adding exponents. When dividing two numbers in scientific notation, you can divide the two significant digit figures and arrive at a power-of-ten by subtracting exponents. Metric notation The metric system, besides being a collection of measurement units for all sorts of physical quantities, is structured around the concept of scientific notation. The primary difference is that the powers-of-ten are represented with alphabetical prefixes instead of by literal powers-of-ten.

The following number line shows some of the more common prefixes and their respective powers-of-ten: Looking at this scale, we can see that 2.5 Gigabytes would mean 2.5 x 109 bytes, or 2.5 billion bytes. Likewise, 3.21 picoamps would mean 3.21 x 10-12 amps, or 3.21 1/trillionths of an amp. Other metric prefixes exist to symbolize powers of ten for extremely small and extremely large multipliers. On the extremely small end of the spectrum, femto (f) = 10-15, atto (a) = 10-18, zepto (z) = 10-21, and yocto (y) = 10-24. On the extremely large end of the spectrum, Peta (P) = 1015, Exa (E) = 1018, Zetta (Z) = 1021, and Yotta (Y) = 1024.

Because the major prefixes in the metric system refer to powers of 10 that are multiples of 3 (from "kilo" on up, and from "milli" on down), metric notation differs from regular scientific notation in that the mantissa can be anywhere between 1 and 999, depending on which prefix is chosen. For example, if a laboratory sample weighs 0.000267 grams, scientific notation and metric notation would express it differently: 2.67 x 10-4 grams (scientific notation) 267 µgrams (metric notation) The same figure may also be expressed as 0.267 milligrams (0.267 mg), although it is usually more common to see the significant digits represented as a figure greater than 1. In recent years a new style of metric notation for electric quantities has emerged which seeks to avoid the use of the decimal point. Since decimal points (".") are easily misread and/or "lost" due to poor print quality, quantities such as 4.7 k may be mistaken for 47 k.

The new notation replaces the decimal point with the metric prefix character, so that "4.7 k" is printed instead as "4k7". Our last figure from the prior example, "0.267 m", would be expressed in the new notation as "0m267". REVIEW: The metric system of notation uses alphabetical prefixes to represent certain powers-of-ten instead of the lengthier scientific notation. Metric prefix conversions To express a quantity in a different metric prefix that what it was originally given, all we need to do is skip the decimal point to the right or to the left as needed. Notice that the metric prefix "number line" in the previous section was laid out from larger to smaller, left to right. This layout was purposely chosen to make it easier to remember which direction you need to skip the decimal point for any given conversion. Example problem: express 0.000023 amps in terms of microamps.

0.000023 amps (has no prefix, just plain unit of amps) From UNITS to micro on the number line is 6 places (powers of ten) to the right, so we need to skip the decimal point 6 places to the right: 0.000023 amps = 23. , or 23 microamps (µA) Example problem: express 304,212 volts in terms of kilovolts. 304,212 volts (has no prefix, just plain unit of volts) From the (none) place to kilo place on the number line is 3 places (powers of ten) to the left, so we need to skip the decimal point 3 places to the left: 304,212. = 304.212 kilovolts (kV) Example problem: express 50.3 Mega-ohms in terms of milli-ohms.

50.3 M ohms (mega = 106) From mega to milli is 9 places (powers of ten) to the right (from 10 to the 6th power to 10 to the -3rd power), so we need to skip the decimal point 9 places to the right: 50.3 M ohms = 50,300,000,000 milli-ohms (mΩ) REVIEW: Follow the metric prefix number line to know which direction you skip the decimal point for conversion purposes. A number with no decimal point shown has an implicit decimal point to the immediate right of the furthest right digit (i.e. for the number 436 the decimal point is to the right of the 6, as such: 436.) Hand calculator use To enter numbers in scientific notation into a hand calculator, there is usually a button marked "E" or "EE" used to enter the correct power of ten.

For example, to enter the mass of a proton in grams (1.67 x 10-24 grams) into a hand calculator, I would enter the following keystrokes: [1] [.] [6] [7] [EE] [2] [4] [+/-] The [+/-] keystroke changes the sign of the power (24) into a -24. Some calculators allow the use of the subtraction key [-] to do this, but I prefer the "change sign" [+/-] key because its more consistent with the use of that key in other contexts. If I wanted to enter a negative number in scientific notation into a hand calculator, I would have to be careful how I used the [+/-] key, lest I change the sign of the power and not the significant digit value.

Pay attention to this example: Number to be entered: -3.221 x 10-15: [3] [.] [2] [2] [1] [+/-] [EE] [1] [5] [+/-] The first [+/-] keystroke changes the entry from 3.221 to -3.221; the second [+/-] keystroke changes the power from 15 to -15. Displaying metric and scientific notation on a hand calculator is a different matter. It involves changing the display option from the normal "fixed" decimal point mode to the "scientific" or "engineering" mode. Your calculator manual will tell you how to set each display mode. These display modes tell the calculator how to represent any number on the numerical readout. The actual value of the number is not affected in any way by the choice of display modes -- only how the number appears to the calculator user. Likewise, the procedure for entering numbers into the calculator does not change with different display modes either. Powers of ten are usually represented by a pair of digits in the upper-right hand corner of the display, and are visible only in the "scientific" and "engineering" modes.

The difference between "scientific" and "engineering" display modes is the difference between scientific and metric notation. In "scientific" mode, the power-of-ten display is set so that the main number on the display is always a value between 1 and 10 (or -1 and -10 for negative numbers). In "engineering" mode, the powers-of-ten are set to display in multiples of 3, to represent the major metric prefixes. All the user has to do is memorize a few prefix/power combinations, and his or her calculator will be "speaking" metric! POWER METRIC PREFIX ----- ------------- 12 ......... Tera (T) 9 .......... Giga (G) 6 .......... Mega (M) 3 .......... Kilo (k) 0 .......... UNITS (plain) -3 ......... milli (m) -6 ......... micro (u) -9 ......... nano (n) -12 ........ pico (p) REVIEW: Use the [EE] key to enter powers of ten. Use "scientific" or "engineering" to display powers of ten, in scientific or metric notation, respectively.

Scientific notation in SPICE The SPICE circuit simulation computer program uses scientific notation to display its output information, and can interpret both scientific notation and metric prefixes in the circuit description files. If you are going to be able to successfully interpret the SPICE analyses throughout this book, you must be able to understand the notation used to express variables of voltage, current, etc. in the program. Let's start with a very simple circuit composed of one voltage source (a battery) and one resistor: To simulate this circuit using SPICE, we first have to designate node numbers for all the distinct points in the circuit, then list the components along with their respective node numbers so the computer knows which component is connected to which, and how. For a circuit of this simplicity, the use of SPICE seems like overkill, but it serves the purpose of demonstrating practical use of scientific notation: Typing out a circuit description file, or netlist, for this circuit, we get this: simple circuit v1 1 0 dc 24 r1 1 0 5 .end The line "v1 1 0 dc 24" describes the battery, positioned between nodes 1 and 0, with a DC voltage of 24 volts.

The line "r1 1 0 5" describes the 5 Ω resistor placed between nodes 1 and 0. Using a computer to run a SPICE analysis on this circuit description file, we get the following results: node voltage ( 1) 24.0000 voltage source currents name current v1 -4.800E+00 total power dissipation 1.15E+02 watts SPICE tells us that the voltage "at" node number 1 (actually, this means the voltage between nodes 1 and 0, node 0 being the default reference point for all voltage measurements) is equal to 24 volts. The current through battery "v1" is displayed as -4.800E+00 amps. This is SPICE's method of denoting scientific notation. What its really saying is "-4.800 x 100 amps," or simply -4.800 amps. The negative value for current here is due to a quirk in SPICE and does not indicate anything significant about the circuit itself.

The "total power dissipation" is given to us as 1.15E+02 watts, which means "1.15 x 102 watts," or 115 watts. Let's modify our example circuit so that it has a 5 kΩ (5 kilo-ohm, or 5,000 ohm) resistor instead of a 5 Ω resistor and see what happens. Once again is our circuit description file, or "netlist:" simple circuit v1 1 0 dc 24 r1 1 0 5k .end The letter "k" following the number 5 on the resistor's line tells SPICE that it is a figure of 5 kΩ, not 5 Ω. Let's see what result we get when we run this through the computer: node voltage ( 1) 24.0000 voltage source currents name current v1 -4.800E-03 total power dissipation 1.15E-01 watts The battery voltage, of course, hasn't changed since the first simulation: its still at 24 volts.

The circuit current, on the other hand, is much less this time because we've made the resistor a larger value, making it more difficult for electrons to flow. SPICE tells us that the current this time is equal to -4.800E-03 amps, or -4.800 x 10-3 amps. This is equivalent to taking the number -4.8 and skipping the decimal point three places to the left. Of course, if we recognize that 10-3 is the same as the metric prefix "milli," we could write the figure as -4.8 milliamps, or -4.8 mA. Looking at the "total power dissipation" given to us by SPICE on this second simulation, we see that it is 1.15E-01 watts, or 1.15 x 10-1 watts. The power of -1 corresponds to the metric prefix "deci," but generally we limit our use of metric prefixes in electronics to those associated with powers of ten that are multiples of three (ten to the power of . . .

-12, -9, -6, -3, 3, 6, 9, 12, etc.). So, if we want to follow this convention, we must express this power dissipation figure as 0.115 watts or 115 milliwatts (115 mW) rather than 1.15 deciwatts (1.15 dW). Perhaps the easiest way to convert a figure from scientific notation to common metric prefixes is with a scientific calculator set to the "engineering" or "metric" display mode. Just set the calculator for that display mode, type any scientific notation figure into it using the proper keystrokes (see your owner's manual), press the "equals" or "enter" key, and it should display the same figure in engineering/metric notation. Again, I'll be using SPICE as a method of demonstrating circuit concepts throughout this book. Consequently, it is in your best interest to understand scientific notation so you can easily comprehend its output data format. Contributors to this chapter are listed in chronological order of their contributions, from most recent to first. See Appendix 2 (Contributor List) for dates and contact information.

Jason Starck (June 2000): HTML document formatting, which led to a much better-looking second edition. Lessons In Electric Circuits copyright (C) 2000-2015 Tony R. Kuphaldt, under the terms and conditions of the Design Science License . Readings on Powers of Ten Basic Math, Scientific Notation, and Astronomical Dimensions Dealing with Numbers Great and Small Peter Wehinger (with an excerpt from Seeing the Light*) People find it cumbersome to speak of units of measurement like "one ten thousandth of a billionth" of an inch, or even "ten to the minus thirteenth" (10-13) inches; so they devise different names. No one gives the distance between cities in inches; we use miles. For people's heights we use feet and inches. But there are 12 inches in a foot, 5280 feet in a mile and so on. That gets messy if we want to convert from miles to inches.

It is easier to use the metric system where everything goes by powers of 10: 1 meter = 1 m = 1 yard 1 kilometer = 1 km = 103 m ~ 5/8 mile 1 centimeter = 1 cm = 10-2 m ~ 2/5 inch (width of a pen) 1 millimeter = 1 mm = 10-3 m ~ 1/25 inch (width of lead in pencil) This much is fine for everyday lengths, but for light we have to use even smaller units of length because its wavelength is so small. Unfortunately, many different names are used. The wavelength of yellow light in vacuum may be called 5750 Å or 575 m or 575 nm or 0.575 m.

The units used here are: 1 Å = 1 Ångstrom = 10-10 m (This unit is often used because a typical atom is a few Ångstroms in size.) 1 nm = 1 nanometer = 10-9 m (formerly written 1 m ) 1 m = 1 micrometer = 1 micron = 10-6 m (This is a convenient unit for high-power microscopes, with which one can look at objects as small as a few micrometers. is the Greek letter "mu") There is a method to the prefixes centi-, milli-, etc. They multiply the unit by some power of 10. For example, milli- always means one thousandth.

The important prefixes are: n = nano- (Greek "dwarf") = 10-9 (thousand millionth) = micro- (Greek "small") = 10-6 (millionth) m = milli- (Latin "thousand") = 10-3 (thousandth) k = kilo- (Greek "thousand") = 103 (thousand) M = mega- (Greek "big") = 106 (million) Thus we can now proudly state that 1 nm = 10-3 m. Scientific Notation: indentAstronomy is the subject of very large numbers. Microscopy is the subject of very small numbers. Consequently, scientific notation has been introduced to deal with such quantities.

For example, the distance from the Earth to the Sun is referred to as the Astronomical Unit (AU) and is: 1 AU = 150,000,000 km = 1.5x l08 km = 9.2 x lO7miles indent Another dimension to illustrate this kind of notation is the mass of the Sun expressed in units of kilograms (kg): Mass of the Sun = 1 Solar Mass = 2.0 x 1030 kg indent Note that 2.0 is the coefficient and 30 is the exponent. Not only can one use scientific notation to express large numbers but also very small numbers. For example, the size of an atom is on the order (or about) a few Ångströms (Å) which is defined as: 1 Ångström = 1 Å = 1.0 x 10-10 m = 0.0000000001 meters indent aNote that the negative sign in the exponent means the value is less than one.

indent Here are some alternate forms of notation for the same numerical values: 3.2 x 108 = 32 x 107 = 320 x 106 = 320,000,000 = 320 million 0.0002 = 2 x 10-4=20 x 10-5= 200x10-6 indent Using a pocket calculator with an exponent key, usually labeled EX or EE or EX, the number: 4.3 x 105 is entered as 4.3 EX 5. Examples of Scientific Notation, Metric Prefixes, and Symbols Word

Question: The prefix micro- means a factor of ten to what negative power?
Answer: 6